# 八元数

${\displaystyle e_{0}}$${\displaystyle e_{1}}$${\displaystyle e_{2}}$${\displaystyle e_{3}}$${\displaystyle e_{4}}$${\displaystyle e_{5}}$${\displaystyle e_{6}}$${\displaystyle e_{7}}$
${\displaystyle ijkl}$形式：

1ijkliljlkl

• ${\displaystyle \mathbb {N} }$ 自然數
• ${\displaystyle \mathbb {Z} }$ 整數
• ${\displaystyle \mathbb {Q} }$ 有理數
• ${\displaystyle \mathbb {R} }$ 實數
• ${\displaystyle \mathbb {C} }$ 複數
• ${\displaystyle \mathbb {H} }$ 四元數

${\displaystyle \mathbb {N} \subseteq \mathbb {Z} \subseteq \mathbb {Q} \subseteq \mathbb {R} \subseteq \mathbb {C} }$

## 定义

${\displaystyle i^{2}=j^{2}=k^{2}=l^{2}=-1\,}$

${\displaystyle x=x_{0}+x_{1}\,i+x_{2}\,j+x_{3}\,k+x_{4}\,l+x_{5}\,il+x_{6}\,jl+x_{7}\,kl}$

${\displaystyle i^{2}=j^{2}=k^{2}=l^{2}=(il)^{2}=(jl)^{2}=(kl)^{2}=-1\,}$

${\displaystyle \times }$ ${\displaystyle 1}$ ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle k}$ ${\displaystyle l}$ ${\displaystyle il}$ ${\displaystyle jl}$ ${\displaystyle kl}$
${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle k}$ ${\displaystyle l}$ ${\displaystyle il}$ ${\displaystyle jl}$ ${\displaystyle kl}$
${\displaystyle i}$ ${\displaystyle i}$ ${\displaystyle -1}$ ${\displaystyle k}$ ${\displaystyle -j}$ ${\displaystyle il}$ ${\displaystyle -l}$ ${\displaystyle -kl}$ ${\displaystyle jl}$
${\displaystyle j}$ ${\displaystyle j}$ ${\displaystyle -k}$ ${\displaystyle -1}$ ${\displaystyle i}$ ${\displaystyle jl}$ ${\displaystyle kl}$ ${\displaystyle -l}$ ${\displaystyle -il}$
${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle j}$ ${\displaystyle -i}$ ${\displaystyle -1}$ ${\displaystyle kl}$ ${\displaystyle -jl}$ ${\displaystyle il}$ ${\displaystyle -l}$
${\displaystyle l}$ ${\displaystyle l}$ ${\displaystyle -il}$ ${\displaystyle -jl}$ ${\displaystyle -kl}$ ${\displaystyle -1}$ ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle k}$
${\displaystyle il}$ ${\displaystyle il}$ ${\displaystyle l}$ ${\displaystyle -kl}$ ${\displaystyle jl}$ ${\displaystyle -i}$ ${\displaystyle -1}$ ${\displaystyle -k}$ ${\displaystyle j}$
${\displaystyle jl}$ ${\displaystyle jl}$ ${\displaystyle kl}$ ${\displaystyle l}$ ${\displaystyle -il}$ ${\displaystyle -j}$ ${\displaystyle k}$ ${\displaystyle -1}$ ${\displaystyle -i}$
${\displaystyle kl}$ ${\displaystyle kl}$ ${\displaystyle -jl}$ ${\displaystyle il}$ ${\displaystyle l}$ ${\displaystyle -k}$ ${\displaystyle -j}$ ${\displaystyle i}$ ${\displaystyle -1}$

${\displaystyle \{e_{0},e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},e_{7}\},}$

${\displaystyle x=x_{0}e_{0}+x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}+x_{4}e_{4}+x_{5}e_{5}+x_{6}e_{6}+x_{7}e_{7},\,}$[9]:5

${\displaystyle e_{i}e_{j}}$[11] ${\displaystyle e_{j}}$
${\displaystyle e_{0}}$ ${\displaystyle e_{1}}$ ${\displaystyle e_{2}}$ ${\displaystyle e_{3}}$ ${\displaystyle e_{4}}$ ${\displaystyle e_{5}}$ ${\displaystyle e_{6}}$ ${\displaystyle e_{7}}$
${\displaystyle e_{i}}$ ${\displaystyle e_{0}}$ ${\displaystyle e_{0}}$ ${\displaystyle e_{1}}$ ${\displaystyle e_{2}}$ ${\displaystyle e_{3}}$ ${\displaystyle e_{4}}$ ${\displaystyle e_{5}}$ ${\displaystyle e_{6}}$ ${\displaystyle e_{7}}$
${\displaystyle e_{1}}$ ${\displaystyle e_{1}}$ ${\displaystyle -e_{0}}$ ${\displaystyle e_{3}}$ ${\displaystyle -e_{2}}$ ${\displaystyle e_{5}}$ ${\displaystyle -e_{4}}$ ${\displaystyle -e_{7}}$ ${\displaystyle e_{6}}$
${\displaystyle e_{2}}$ ${\displaystyle e_{2}}$ ${\displaystyle -e_{3}}$ ${\displaystyle -e_{0}}$ ${\displaystyle e_{1}}$ ${\displaystyle e_{6}}$ ${\displaystyle e_{7}}$ ${\displaystyle -e_{4}}$ ${\displaystyle -e_{5}}$
${\displaystyle e_{3}}$ ${\displaystyle e_{3}}$ ${\displaystyle e_{2}}$ ${\displaystyle -e_{1}}$ ${\displaystyle -e_{0}}$ ${\displaystyle e_{7}}$ ${\displaystyle -e_{6}}$ ${\displaystyle e_{5}}$ ${\displaystyle -e_{4}}$
${\displaystyle e_{4}}$ ${\displaystyle e_{4}}$ ${\displaystyle -e_{5}}$ ${\displaystyle -e_{6}}$ ${\displaystyle -e_{7}}$ ${\displaystyle -e_{0}}$ ${\displaystyle e_{1}}$ ${\displaystyle e_{2}}$ ${\displaystyle e_{3}}$
${\displaystyle e_{5}}$ ${\displaystyle e_{5}}$ ${\displaystyle e_{4}}$ ${\displaystyle -e_{7}}$ ${\displaystyle e_{6}}$ ${\displaystyle -e_{1}}$ ${\displaystyle -e_{0}}$ ${\displaystyle -e_{3}}$ ${\displaystyle e_{2}}$
${\displaystyle e_{6}}$ ${\displaystyle e_{6}}$ ${\displaystyle e_{7}}$ ${\displaystyle e_{4}}$ ${\displaystyle -e_{5}}$ ${\displaystyle -e_{2}}$ ${\displaystyle e_{3}}$ ${\displaystyle -e_{0}}$ ${\displaystyle -e_{1}}$
${\displaystyle e_{7}}$ ${\displaystyle e_{7}}$ ${\displaystyle -e_{6}}$ ${\displaystyle e_{5}}$ ${\displaystyle e_{4}}$ ${\displaystyle -e_{3}}$ ${\displaystyle -e_{2}}$ ${\displaystyle e_{1}}$ ${\displaystyle -e_{0}}$

${\displaystyle e_{i}e_{j}={\begin{cases}e_{j},&{\text{if }}i=0\\e_{i},&{\text{if }}j=0\\-\delta _{ij}e_{0}+\varepsilon _{ijk}e_{k},&{\text{otherwise}}\end{cases}}}$

### 凯莱-迪克松构造

${\displaystyle (a,b)(c,d)=(ac-d^{*}b,da+bc^{*})}$

### 法诺平面记忆

(a, b, c)为位于一条给定的直线上的三个有序点，其顺序由箭头的方向指定。那么，乘法由下式给出：[18]

ab = cba = −c

• 1是乘法单位元，
• 对于图中的每一个点，都有${\displaystyle e^{2}=-1}$

### 共轭、範数和逆元素

${\displaystyle x=x_{0}+x_{1}\,i+x_{2}\,j+x_{3}\,k+x_{4}\,l+x_{5}\,il+x_{6}\,jl+x_{7}\,kl}$

${\displaystyle x^{*}=x_{0}-x_{1}\,i-x_{2}\,j-x_{3}\,k-x_{4}\,l-x_{5}\,il-x_{6}\,jl-x_{7}\,kl.}$

${\displaystyle x^{*}={\overline {x}}=x_{0}e_{0}-x_{i}e_{i},\ i=1,2\cdots 7}$

x的实数部分定义为${\displaystyle \mathrm {Re} \left(x\right)={\tfrac {x+x^{*}}{2}}=x_{0}}$，虚数部分定义为${\displaystyle \mathrm {Im} \left(x\right)={\tfrac {x-x^{*}}{2}}}$[16]所有纯虚的八元数生成了${\displaystyle \mathbb {O} }$的一个七维子空间，记为Im(${\displaystyle \mathbb {O} }$)[8]:186

${\displaystyle \|x\|={\sqrt {x^{*}x}}}$

${\displaystyle \|x\|^{2}=x^{*}x=x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}}$

${\displaystyle \mathbb {O} }$上範数的存在，意味着${\displaystyle \mathbb {O} }$的所有非零元素都存在逆元素x ≠ 0的逆元素为：[16][9]:6

${\displaystyle x^{-1}={\frac {x^{*}}{\|x\|^{2}}}}$

## 性质

${\displaystyle ij=-ji\neq ji\,}$

${\displaystyle (ij)l=-i(jl)\neq i(jl)\,}$

${\displaystyle \|xy\|=\|x\|\|y\|}$

### 自同构

${\displaystyle A(xy)=A(x)A(y).\,}$

${\displaystyle \mathbb {O} }$的所有自同构的集合组成了一个，称为G2英语G2 (mathematics)[21][9]G2是一个单连通紧致、14维的实李群[22]这个群是中最小的一个。[23]

## 註釋

1. ^ 在範数可良好定義的前提下，${\displaystyle {\frac {x+x^{*}}{2}}\in \mathbb {R} }$，且${\displaystyle x^{*}x>0}$[16]，因此可以得到${\displaystyle x^{*}x=xx^{*}}$总是非负实数的結論。

## 参考文献

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